ON SPECTRAL METHODS FOR VOLTERRA INTEGRAL EQUATIONS AND THE CONVERGENCE ANALYSIS

The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is proposed to solve the Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical results confirm the theoretical prediction of the exponential rate of convergence. The result in this work seems to be the first successful spectral approach （with theoretical justification） for the Volterra type equations.     

INVERSE COEFFICIENT PROBLEMS FOR PARABOLIC HEMIVARIATIONAL INEQUALITIES

This paper is devoted to the class of inverse problems for a nonlinear parabolic hemivariational inequality. The unknown coefficient of the operator depends on the gradient of the solution and belongs to a set of admissible coefficients. It is proved that the convergence of solutions for the corresponding direct problems continuously depends on the coefficient convergence. Based on this result the existence of a quasisolution of the inverse problem is obtained.     

Minimax Optimal Rates of Convergence for Multicategory Classifications

In the problem of classification （or pattern recognition）, given a set of n samples, we attempt to construct a classifier gn with a small misclassification error. It is important to study the convergence rates of the misclassification error as n tends to infinity. It is known that such a rate can＇t exist for the set of all distributions. In this paper we obtain the optimal convergence rates for a class of distributions L^（λ,ω） in multicategory classification and nonstandard binary classification.     

Convergence and Optimality of Adaptive Regularization for Ill-posed Deconvolution Problems in Infinite Spaces

The adaptive regularization method is first proposed by Ryzhikov et al. in [6] for the deconvolution in elimination of multiples which appear frequently in geoscience and remote sensing. They have done experiments to show that this method is very effective. This method is better than the Tikhonov regularization in the sense that it is adaptive, i.e., it automatically eliminates the small eigenvalues of the operator when the operator is near singular. In this paper, we give theoretical analysis about the adaptive regularization. We introduce an a priori strategy and an a posteriori strategy for choosing the regularization parameter, and prove regularities of the adaptive regularization for both strategies. For the former, we show that the order of the convergence rate can approach O（｜｜n｜｜^4v/4v＋1） for some 0 〈 v 〈 1, while for the latter, the order of the convergence rate can be at most O（｜｜n｜｜^2v/2v＋1） for some 0 〈 v 〈 1.     

A DIRECT SEARCH FRAME-BASED CONJUGATE GRADIENTS METHOD

A derivative-free frame-based conjugate gradients algorithm is presented.Convergenceis shown for C~1 functions,and this is verified in numerical trials.The algorithm is tested ona variety of low dimensional problems,some of which are ill-conditioned,and is also testedon problems of high dimension.Numerical results show that the algorithm is effectiveon both classes of problems.The results are compared with those from a discrete quasi-Newton method,showing that the conjugate gradients algorithm is competitive.Thealgorithm exhibits the conjugate gradients speed-up on problems for which the Hessian atthe solution has repeated or clustered eigenvalues.The algorithm is easily parallelizable.     

A FAST CONVERGENT METHOD OF ITERATED REGULARIZATION

This article presents a fast convergent method of iterated regularization based on the idea of Landweber iterated regularization, and a method for a-posteriori choice by the Morozov discrepancy principle and the optimum asymptotic convergence order of the regularized solution is obtained. Numerical test shows that the method of iterated regularization can quicken the convergence speed and reduce the calculation burden efficiently.     

关于MQ算法的松驰因子

In this paper, a new method, so called A-method, is given for the convergence analysis of the MQ-algorithm. And the finer relaxation parameter θA is obtained. The numerical results show that our new method has the outstanding effect of accelerating convergence. Moreover, the relaxation parameter θA is the optimum in a point of view.     

MIXED FINITE ELEMENT METHOD FOR THE CONVECTION-DOMINATED DIFFUSION PROBLEMS WITH SMALL PARAMETER ε

In rids paper a mixed finite element method for the convection-dominated diffusion problems with small parameter ε is presented,the effect of the parameter ε on the approximation error is considered and a sufficient condition for optimal error estimates is derived. The paper also shows that under some conditions,the standard finite dement method only gives a hounded solution,however the mixed finite element method gives a convergent one.     

LELONG-DEMAILLY NUMBERS IN TERMS OF CAPACITY AND WEAK CONVERGENCE FOR CLOSED POSITIVE CURRENTS

In this paper we give a new definition of the Lelong-Demailly number in terms of the CT-capacity, where T is a closed positive current and CT is the capacity associated to T. This derived from some esimate on the growth of the CT-capacity of the sublevel sets of a weighted plurisubharmonic （psh for short） function. These estimates enable us to give another proof of the Demailly＇s comparaison theorem as well as a generalization of some results due to Xing concerning the characterization of bounded psh functions. Another problem that we consider here is related to the existence of a psh function v that satisfies the equality CT（K） ： fK T ∧ （dd^cu）^p, where K is a compact subset. Finally, we give some conditions on the capacity CT that guarantees the weak convergence ukTk → uT, for positive closed currents T, Tk and psh functions uk, u.     

CONVERGENCE RATE OF A GENERALIZED ADDITIVE SCHWARZ ALGORITHM

The convergence rate of a generalized additive Schwarz algorithm for solving boundaryvalue problems of elliptic partial differential equations is studied.A quantitative analysisof the convergence rate is given for the model Dirichlet problem.It will be shown that agreater acceleration of the algorithm can be obtained by choosing the parameter suitably.Some numerical tests are also presented in this paper.     

A new modified one-step smoothing Newton method for solving the general mixed complementarity problem

In last decades, there has been much effort on the solution and the analysis of the mixed complementarity problem (MCP) by reformulating MCP as an unconstrained minimization involving an MCP function. In this paper, we propose a new modified one-step smoothing Newton method for solving general (not necessarily P₀) mixed complementarity problems based on well-known Chen-Harker-Kanzow-Smale smooth function. Under suitable assumptions, global convergence and locally superlinear convergence of the algorithm are established.     

One-step smoothing Newton method for solving the mixed complementarity problem with a function

The mixed complementarity problem (denote by MCP(F)) can be reformulated as the solution of a smooth system of equations. In the paper, based on a perturbed mid function, we propose a new smoothing function, which has an important property, not satisfied by many other smoothing function. The existence and continuity of a smooth path for solving the mixed complementarity problem with a P₀ function are discussed. Then we presented a one-step smoothing Newton algorithm to solve the MCP with a P₀ function. The global convergence of the proposed algorithm is verified under mild conditions. And by using the smooth and semismooth technique, the rate of convergence of the method is proved under some suitable assumptions.     

关于解极大相关问题P-SOR算法的收敛性

对求解极大相关问题的P-SOR方法的收敛性做了进一步研究.得到了一些新的收敛条件.为了提高收敛到全局最大解的可能性,提出了一种新的初始向量选择策略.给出了P-SOR算法的对称形式(P-SSOR).还给出了一种算法精化策略.最后,用数值例子说明新方法的有效性.     

On convergence of iterative methods for maximal correlation problems

Several iterative methods for maximal correlation problems (MCPs) have been proposed in the literature. This paper deals with the convergence of these iterations and contains three contributions. Firstly, a unified and concise proof of the monotone convergence of these iterative methods is presented. Secondly, a starting point strategy is analysed. Thirdly, some error estimates are presented to test the quality of a computed solution. Both theoretical results and numerical tests suggest that combining with this starting point strategy these methods converge rapidly and are more likely converging to a global maximizer of MCP. Copyright © 2016 John Wiley & Sons, Ltd.     

Riemannian Trust-Region Method for the Maximal Correlation Problem

The maximal correlation problem (MCP) arising in the canonical correlation analysis is very important to assess the relationship between sets of random variables. Efficient and fast methods for solving MCP are desired in broad statistical and nonstatistical applications. Some early proposed algorithms are based on the first-order information of MCP, and fast convergence could not be expected. In this article, we turn the generic Riemannian trust-region method of Absil et al. [2 P.-A. Absil , C. G. Baker , and K. A. Gallivan ( 2007 ). Trust-region methods on Riemannian manifolds . Found. Comput. Math. 7 : 303 – 330 . [Google Scholar]] into a practical algorithm for MCP, which enjoys the global convergence and local superlinear convergence rate. The structure-exploiting preconditioning technique is also discussed in solving the trust-region subproblem. Numerical empirical evaluation and a comparison against other methods are reported, which shows that the method is efficient in solving MCPs.     

A class of smoothing functions for nonlinear and mixed complementarity problems

We propose a class of parametric smooth functions that approximate the fundamental plus function, (x)₊=max{0, x}, by twice integrating a probability density function. This leads to classes of smooth parametric nonlinear equation approximations of nonlinear and mixed complementarity problems (NCPs and MCPs). For any solvable NCP or MCP, existence of an arbitrarily accurate solution to the smooth nonlinear equations as well as the NCP or MCP, is established for sufficiently large value of a smoothing parameter . Newton-based algorithms are proposed for the smooth problem. For strongly monotone NCPs, global convergence and local quadratic convergence are established. For solvable monotone NCPs, each accumulation point of the proposed algorithms solves the smooth problem. Exact solutions of our smooth nonlinear equation for various values of the parameter , generate an interior path, which is different from the central path for interior point method. Computational results for 52 test problems compare favorably with these for another Newton-based method. The smooth technique is capable of solving efficiently the test problems solved by Dirkse and Ferris [6], Harker and Xiao [11] and Pang & Gabriel [28].This material is based on research supported by Air Force Office of Scientific Research Grant F49620-94-1-0036 and National Science Foundation Grant CCR-9322479.     

Strictly feasible equation-based methods for mixed complementarity problems

Summary.   We introduce a new algorithm for the solution of the mixed complementarity problem (MCP) which has stronger properties than most existing methods. In fact, typical solution methods for the MCP either generate feasible iterates but have to solve relatively complicated subproblems (like quadratic programs or linear complementarity problems), or they have relatively simple subproblems (like linear systems of equations) but generate not necessarily feasible iterates. The method to be presented here combines the nice features of these two classes of methods: It has to solve only one linear system of equations (of reduced dimension) at each iteration, and it generates feasible (more precisely: strictly feasible) iterates. The new method has some nice global and local convergence properties. Some preliminary numerical results will also be given. Received August 26, 1999 / Revised version recived April 11, 2000 / Published online February 5, 2001     

An alternating variable method for the maximal correlation problem

The maximal correlation problem (MCP) aiming at optimizing correlations between sets of variables plays an important role in many areas of statistical applications. Up to date, algorithms for the general MCP stop at solutions of the multivariate eigenvalue problem (MEP), which serves only as a necessary condition for the global maxima of the MCP. For statistical applications, the global maximizer is quite desirable. In searching the global solution of the MCP, in this paper, we propose an alternating variable method (AVM), which contains a core engine in seeking a global maximizer. We prove that (i) the algorithm converges globally and monotonically to a solution of the MEP, (ii) any convergent point satisfies a global optimal condition of the MCP, and (iii) whenever the involved matrix A is nonnegative irreducible, it converges globally to the global maximizer. These properties imply that the AVM is an effective approach to obtain a global maximizer of the MCP. Numerical testings are carried out and suggest a superior performance to the others, especially in finding a global solution of the MCP.     

An extension of the Jacobi algorithm for multi-valued mixed complementarity problems

We consider a generalized mixed complementarity problem (MCP) with box constraints and multi-valued cost mapping. We introduce a concept of an upper Z-mapping, which generalizes the well-known concept of the single-valued Z-mapping and involves the diagonal multi-valued mappings, and suggest an extension of the Jacobi algorithm for the above problem containing a composition of such mappings. Being based on its convergence theorem, we establish several existence and uniqueness results. Some examples of the applications are also given.     

An Alternating Direction Method of Multipliers for MCP-penalized Regression with High-dimensional Data

The minimax concave penalty (MCP) has been demonstrated theoretically and practically to be effective in nonconvex penalization for variable selection and parameter estimation. In this paper, we develop an efficient alternating direction method of multipliers (ADMM) with continuation algorithm for solving the MCP-penalized least squares problem in high dimensions. Under some mild conditions, we study the convergence properties and the Karush–Kuhn–Tucker (KKT) optimality conditions of the proposed method. A high-dimensional BIC is developed to select the optimal tuning parameters. Simulations and a real data example are presented to illustrate the efficiency and accuracy of the proposed method.